Diwali, the festival of light celebrated across the world, is falling on 30th/10 this year, and I tried a way to link up the date part (3 of 30) and the celebrations that go with it by drawing the image displayed below which can be used as a mod 3 graph!
Start at node 0, for each digit D of the number, move along D black arrows (for digit 0 do not move at all), and as you pass from one digit to the next, move along a single white arrow.
For example, let n = 325, Start at node 0, move along 3 black arrows(to node 0), then 1 white arrow(to node 0), then since the next digit is 2 move along 2 black arrows (to node 2), then 1 white arrow (to node 2), and finally 5 black arrows(to node 1). Finishing at node 1 shows that the remainder on dividing 325 is 1.
If you try this for a number that is divisible by 3, say 9234, you will always end up in node 0. Therefore, this "div-a" can be used to test divisibility by 3. I want to know wether it's necessary to have the three yellow loops(representing flames) or can I do away with them all together and still perform mod 3 .(BTW : Div-a in Hindi means earthen lamp)
You can do away with it, with or without the yellow loop,which is a self loop ,you end up with the same solution.
As why does the method work:
Notice that $$10 \equiv 1 \mod 3$$
and hence $$\sum_{k} a_k10^k \equiv \sum_{k} a_k \mod 3$$